LV


Casos brutos (1116 municípios)


# A tibble: 6 × 13
  year   mean    sd   var   min quant25 median quant75 quant90 quant95   max  zero zero.perc
  <chr> <dbl> <dbl> <dbl> <dbl>   <dbl>  <dbl>   <dbl>   <dbl>   <dbl> <dbl> <int>     <dbl>
1 2001   2.56  11.8  139.     0       0      0       1     4       7     173   645     0.578
2 2002   2.49  12.3  151.     0       0      0       1     4       8     214   702     0.629
3 2003   3.00  15.2  232.     0       0      0       1     4.5     9     303   654     0.586
4 2004   3.38  19.4  376.     0       0      0       1     5      12     464   692     0.620
5 2005   3.45  18.4  338.     0       0      0       1     5.5    12.2   412   662     0.593
6 2006   3.52  17.9  319.     0       0      0       1     5.5    12.2   302   661     0.592

Brasil (whole)


# A tibble: 5 × 13
   year      mean      sd     var   min quant25 median quant75     max  zero zero.perc  `NA` NA.perc
  <dbl>     <dbl>   <dbl>   <dbl> <dbl>   <dbl>  <dbl>   <dbl>   <dbl> <int>     <dbl> <int>   <dbl>
1  2001   1.05e-5 5.52e-5 3.05e-9     0       0      0 5.87e-5 0.00120  5089     0.913    12 0.00215
2  2002   8.95e-6 5.44e-5 2.96e-9     0       0      0 4.12e-5 0.00128  5146     0.924    12 0.00215
3  2003   1.09e-5 6.11e-5 3.73e-9     0       0      0 5.66e-5 0.00167  5098     0.915    12 0.00215
4  2004   9.90e-6 5.74e-5 3.29e-9     0       0      0 4.83e-5 0.00173  5140     0.922     8 0.00144
5  2005   1.04e-5 5.67e-5 3.22e-9     0       0      0 5.62e-5 0.00131  5110     0.917     8 0.00144

LTA


Casos brutos (3212 municípios)


# A tibble: 6 × 13
  year   mean    sd   var   min quant25 median quant75 quant90 quant95   max  zero zero.perc
  <chr> <dbl> <dbl> <dbl> <dbl>   <dbl>  <dbl>   <dbl>   <dbl>   <dbl> <dbl> <int>     <dbl>
1 2001  10.6   45.0 2025.     0       0      1       6      23    50    1772  1289     0.401
2 2002   9.66  33.8 1141.     0       0      1       6      23    49.4  1144  1224     0.381
3 2003  10.1   46.9 2196.     0       0      1       6      24    49    2107  1294     0.403
4 2004   9.28  34.9 1215.     0       0      1       5      20    43    1023  1402     0.436
5 2005   8.62  29.5  869.     0       0      1       5      20    44.4   928  1271     0.396
6 2006   6.93  23.9  570.     0       0      1       4      16    34     720  1378     0.429

Brasil (whole)


# A tibble: 5 × 13
   year     mean       sd      var   min quant25 median quant75    max  zero zero.perc  `NA` NA.perc
  <dbl>    <dbl>    <dbl>    <dbl> <dbl>   <dbl>  <dbl>   <dbl>  <dbl> <int>     <dbl> <int>   <dbl>
1  2001 0.000293 0.00122   1.49e-6     0       0      0 0.00157 0.0416  3637     0.653    12 0.00215
2  2002 0.000284 0.00104   1.09e-6     0       0      0 0.00151 0.0225  3572     0.641    12 0.00215
3  2003 0.000287 0.00107   1.16e-6     0       0      0 0.00160 0.0287  3642     0.654    12 0.00215
4  2004 0.000242 0.000924  8.53e-7     0       0      0 0.00130 0.0201  3754     0.674     8 0.00144
5  2005 0.000241 0.000851  7.24e-7     0       0      0 0.00132 0.0149  3623     0.650     8 0.00144

UR


# A tibble: 5 × 9
   year  mean    sd   var   min quant25 median quant75   max
  <int> <dbl> <dbl> <dbl> <dbl>   <dbl>  <dbl>   <dbl> <dbl>
1  2001  74.5  7.94  63.1  59.3    67.4   74.6    81.2  90.1
2  2002  74.5  7.67  58.9  59.0    67.7   74.4    81.1  90.0
3  2003  74.6  7.96  63.4  59.2    67.7   74.6    81.5  90.1
4  2004  74.6  7.79  60.7  58.8    67.8   74.6    81.3  89.6
5  2005  73.6  8.21  67.4  58.7    66.5   73.8    80.5  89.8

SDII


# A tibble: 5 × 9
   year  mean    sd   var   min quant25 median quant75   max
  <int> <dbl> <dbl> <dbl> <dbl>   <dbl>  <dbl>   <dbl> <dbl>
1  2001  4.65 0.855 0.730  2.4     4.18   4.59    5.03  8.72
2  2002  4.46 0.797 0.636  2.48    4.02   4.42    4.82  8.98
3  2003  4.64 0.787 0.619  2.45    4.22   4.57    4.99  9.08
4  2004  4.69 0.738 0.545  2.82    4.27   4.58    5.04  8.68
5  2005  4.62 0.823 0.677  2.6     4.12   4.47    5     9.48

TASMAX


# A tibble: 5 × 9
   year  mean    sd   var   min quant25 median quant75   max
  <int> <dbl> <dbl> <dbl> <dbl>   <dbl>  <dbl>   <dbl> <dbl>
1  2001  27.5  2.89  8.33  20.1    25.4   27.4    29.8  33.8
2  2002  27.5  2.98  8.89  20.2    25.1   27.4    29.9  33.9
3  2003  27.6  2.93  8.61  20.3    25.3   27.5    30.0  34.1
4  2004  27.5  3.06  9.37  19.8    25.2   27.4    29.8  33.9
5  2005  28.0  2.93  8.60  20.3    25.8   27.8    30.4  33.9

TASMIN


# A tibble: 5 × 9
   year  mean    sd   var   min quant25 median quant75   max
  <int> <dbl> <dbl> <dbl> <dbl>   <dbl>  <dbl>   <dbl> <dbl>
1  2001  17.7  2.97  8.85  10.9    15.2   17.8    20.2  25.4
2  2002  17.7  3.00  8.99  10.9    15.2   17.8    20.2  25.4
3  2003  17.9  2.93  8.61  11.2    15.4   18.0    20.3  25.4
4  2004  17.8  3.05  9.33  10.9    15.2   17.9    20.3  25.6
5  2005  17.9  2.96  8.78  11.1    15.5   18.1    20.4  25.6

PREC_MEAN


# A tibble: 5 × 9
   year  mean    sd   var   min quant25 median quant75   max
  <int> <dbl> <dbl> <dbl> <dbl>   <dbl>  <dbl>   <dbl> <dbl>
1  2001  4.42 0.914 0.836  1.99    3.93   4.27    4.86  8.72
2  2002  4.21 0.847 0.717  2.23    3.74   4.08    4.60  8.96
3  2003  4.40 0.854 0.729  2.16    3.90   4.26    4.81  8.98
4  2004  4.44 0.811 0.658  2.52    3.95   4.28    4.86  8.65
5  2005  4.37 0.889 0.790  2.34    3.84   4.17    4.83  9.40

PREC_SUM


# A tibble: 5 × 9
   year  mean    sd   var   min quant25 median quant75   max
  <int> <dbl> <dbl> <dbl> <dbl>   <dbl>  <dbl>   <dbl> <dbl>
1  2001  134.  27.7  770.  60.2    119.   129.    147.  264.
2  2002  128.  25.7  662.  67.2    113.   124.    139.  271.
3  2003  133.  25.9  673.  65.0    118.   129.    146.  272.
4  2004  135.  24.7  609.  77.0    120.   130.    148.  264.
5  2005  133.  27.0  727.  70.7    116.   126.    147.  285.

Modelagem


PREC_MEAN e PREC_SUM são “super” colineares / dependentes / redundantes / dizem a mesma coisa. Usaremos apenas uma, a PREC_SUM.

Como estamos trabalhando com incidências, não contagens absolutas, modelos com inflação de zero não são viáveis. Pela presença de zeros, distribuições não-simétricas para dados contínuos, como a Gama e a Normal Inversa, também não são viáveis. Sendo assim, ficamos com a distribuição Normal.

Ajustamos modelos mistos / multiníveis / hierárquicos à nível de município, para assim acomodar a dependência das cinco observações de cada. Tentamos também modelar essa dependência intra-município de um modo temporal, mas não obtivemos convergência numérica. Sendo assim, acomodamos o efeito dos anos no efeito fixo, junto das variáveis climáticas.

Para as variáveis estatisticamente significativas dentro de cada bioma, gráficos são fornecidos para a partir deles limiares serem obtidos.

Modelando LV por bioma


Bioma Amazonia + Pantanal


# A tibble: 5 × 12
   year       mean        sd          var   min quant25 median quant75     max     n  zero zero.perc
  <dbl>      <dbl>     <dbl>        <dbl> <dbl>   <dbl>  <dbl>   <dbl>   <dbl> <int> <int>     <dbl>
1  2001 0.00000954 0.0000520      2.71e-9     0       0      0       0 5.84e-4   554   508     0.917
2  2002 0.0000102  0.0000625      3.91e-9     0       0      0       0 8.88e-4   554   508     0.917
3  2003 0.0000125  0.0000703      4.94e-9     0       0      0       0 8.08e-4   554   511     0.922
4  2004 0.0000142  0.0000711      5.05e-9     0       0      0       0 6.32e-4   554   503     0.908
5  2005 0.0000192  0.000102       1.04e-8     0       0      0       0 1.31e-3   554   501     0.904

# A tibble: 7 × 4
  var              Estimate `Std. Error`   p.value
  <chr>               <dbl>        <dbl>     <dbl>
1 (Intercept) -0.00462      0.00122      NA       
2 year         0.00000229   0.000000611   0.000185
3 UR           0.000000115  0.000000212   0.589   
4 SDII         0.000000865  0.00000121    0.475   
5 TASMAX       0.000000862  0.000000906   0.341   
6 TASMIN       0.000000606  0.000000863   0.483   
7 PREC_SUM     0.0000000125 0.0000000380  0.741   

Bioma Caatinga


# A tibble: 5 × 12
   year      mean        sd           var   min quant25 median quant75     max     n  zero zero.perc
  <dbl>     <dbl>     <dbl>         <dbl> <dbl>   <dbl>  <dbl>   <dbl>   <dbl> <int> <int>     <dbl>
1  2001 0.0000273 0.0000830 0.00000000688     0       0      0       0 1.13e-3  1079   847     0.785
2  2002 0.0000193 0.0000617 0.00000000381     0       0      0       0 6.76e-4  1079   884     0.819
3  2003 0.0000243 0.0000823 0.00000000678     0       0      0       0 1.24e-3  1079   873     0.809
4  2004 0.0000232 0.0000791 0.00000000626     0       0      0       0 9.94e-4  1079   888     0.823
5  2005 0.0000261 0.0000800 0.00000000640     0       0      0       0 1.11e-3  1079   875     0.811

# A tibble: 7 × 4
  var             Estimate `Std. Error`     p.value
  <chr>              <dbl>        <dbl>       <dbl>
1 (Intercept) -0.00191     0.00122      NA         
2 year         0.000000925 0.000000607   0.128     
3 UR           0.000000729 0.000000186   0.0000924 
4 SDII        -0.00000621  0.00000136    0.00000529
5 TASMAX       0.00000153  0.000000693   0.0269    
6 TASMIN       0.00000171  0.000000612   0.00520   
7 PREC_SUM    -0.000000123 0.0000000479  0.0105    

Bioma Cerrado


# A tibble: 5 × 12
   year      mean        sd           var   min quant25 median quant75     max     n  zero zero.perc
  <dbl>     <dbl>     <dbl>         <dbl> <dbl>   <dbl>  <dbl>   <dbl>   <dbl> <int> <int>     <dbl>
1  2001 0.0000148 0.0000724 0.00000000525     0       0      0       0 1.20e-3  1070   965     0.902
2  2002 0.0000160 0.0000891 0.00000000795     0       0      0       0 1.28e-3  1070   970     0.907
3  2003 0.0000190 0.0000875 0.00000000766     0       0      0       0 1.67e-3  1070   931     0.870
4  2004 0.0000141 0.0000650 0.00000000423     0       0      0       0 8.63e-4  1070   962     0.899
5  2005 0.0000119 0.0000507 0.00000000257     0       0      0       0 7.46e-4  1070   956     0.893

# A tibble: 7 × 4
  var              Estimate `Std. Error`  p.value
  <chr>               <dbl>        <dbl>    <dbl>
1 (Intercept)  0.00274      0.00106      NA      
2 year        -0.00000140   0.000000530   0.00823
3 UR           0.000000332  0.000000178   0.0629 
4 SDII         0.00000178   0.00000115    0.122  
5 TASMAX       0.000000890  0.000000639   0.164  
6 TASMIN       0.000000869  0.000000551   0.114  
7 PREC_SUM     0.0000000422 0.0000000393  0.282  

Bioma Mata Atlantica


# A tibble: 5 × 12
   year       mean        sd      var   min quant25 median quant75      max     n  zero zero.perc
  <dbl>      <dbl>     <dbl>    <dbl> <dbl>   <dbl>  <dbl>   <dbl>    <dbl> <int> <int>     <dbl>
1  2001 0.00000291 0.0000263 6.93e-10     0       0      0       0 0.000890  2751  2654     0.965
2  2002 0.00000213 0.0000220 4.85e-10     0       0      0       0 0.000598  2751  2669     0.970
3  2003 0.00000247 0.0000265 7.05e-10     0       0      0       0 0.000678  2751  2669     0.970
4  2004 0.00000254 0.0000375 1.40e- 9     0       0      0       0 0.00173   2751  2672     0.971
5  2005 0.00000223 0.0000267 7.14e-10     0       0      0       0 0.00112   2751  2663     0.968

# A tibble: 7 × 4
  var         Estimate `Std. Error`  p.value
  <chr>          <dbl>        <dbl>    <dbl>
1 (Intercept)  4.27e-4 0.000289     NA      
2 year        -2.12e-7 0.000000145   0.144  
3 UR           6.91e-9 0.0000000472  0.884  
4 SDII        -1.16e-6 0.000000357   0.00117
5 TASMAX       2.10e-8 0.000000160   0.896  
6 TASMIN       2.88e-7 0.000000126   0.0219 
7 PREC_SUM    -8.46e-9 0.0000000105  0.419  

Bioma Pampa


# A tibble: 5 × 12
   year       mean        sd      var   min quant25 median quant75      max     n  zero zero.perc
  <dbl>      <dbl>     <dbl>    <dbl> <dbl>   <dbl>  <dbl>   <dbl>    <dbl> <int> <int>     <dbl>
1  2001 0.00000183 0.0000197 3.88e-10     0       0      0       0 0.000212   118   115     0.975
2  2002 0.00000274 0.0000296 8.73e-10     0       0      0       0 0.000318   118   115     0.975
3  2003 0.00000367 0.0000394 1.55e- 9     0       0      0       0 0.000425   118   114     0.966
4  2004 0.00000137 0.0000148 2.19e-10     0       0      0       0 0.000159   118   115     0.975
5  2005 0.00000229 0.0000247 6.08e-10     0       0      0       0 0.000266   118   115     0.975

# A tibble: 7 × 4
  var              Estimate `Std. Error`  p.value
  <chr>               <dbl>        <dbl>    <dbl>
1 (Intercept)  0.000231     0.000667     NA      
2 year        -0.000000146  0.000000336   0.665  
3 UR           0.000000550  0.000000187   0.00428
4 SDII        -0.00000361   0.00000135    0.00870
5 TASMAX       0.00000156   0.000000541   0.00493
6 TASMIN      -0.000000141  0.000000314   0.655  
7 PREC_SUM     0.0000000139 0.0000000326  0.671  

Modelando LTA por bioma


Bioma Amazonia + Pantanal


# A tibble: 5 × 12
   year     mean      sd        var   min quant25     median  quant75    max     n  zero zero.perc
  <dbl>    <dbl>   <dbl>      <dbl> <dbl>   <dbl>      <dbl>    <dbl>  <dbl> <int> <int>     <dbl>
1  2001 0.000364 0.00102 0.00000104     0       0 0.00000772 0.000252 0.0128   554   268     0.484
2  2002 0.000542 0.00147 0.00000215     0       0 0.00000426 0.000362 0.0138   554   275     0.496
3  2003 0.000883 0.00212 0.00000449     0       0 0.0000315  0.000815 0.0287   554   255     0.460
4  2004 0.000918 0.00201 0.00000405     0       0 0.0000772  0.000820 0.0191   554   234     0.422
5  2005 0.000792 0.00171 0.00000294     0       0 0.0000744  0.000617 0.0149   554   228     0.412

# A tibble: 7 × 4
  var            Estimate `Std. Error`   p.value
  <chr>             <dbl>        <dbl>     <dbl>
1 (Intercept) -0.241       0.0318      NA       
2 year         0.000119    0.0000159    8.49e-14
3 UR           0.0000123   0.00000537   2.23e- 2
4 SDII         0.0000330   0.0000308    2.85e- 1
5 TASMAX       0.0000246   0.0000226    2.77e- 1
6 TASMIN       0.0000338   0.0000218    1.22e- 1
7 PREC_SUM    -0.00000114  0.000000976  2.41e- 1

Bioma Caatinga


# A tibble: 5 × 12
   year     mean       sd         var   min quant25    median  quant75    max     n  zero zero.perc
  <dbl>    <dbl>    <dbl>       <dbl> <dbl>   <dbl>     <dbl>    <dbl>  <dbl> <int> <int>     <dbl>
1  2001 0.000752 0.00201  0.00000403      0       0 0.0000598 0.000746 0.0416  1079   506     0.469
2  2002 0.000683 0.00148  0.00000220      0       0 0.0000561 0.000722 0.0180  1079   485     0.449
3  2003 0.000435 0.00119  0.00000142      0       0 0         0.000366 0.0172  1079   566     0.525
4  2004 0.000384 0.00116  0.00000135      0       0 0         0.000323 0.0201  1079   561     0.520
5  2005 0.000350 0.000825 0.000000681     0       0 0         0.000342 0.0128  1079   543     0.503

# A tibble: 7 × 4
  var             Estimate `Std. Error`   p.value
  <chr>              <dbl>        <dbl>     <dbl>
1 (Intercept)  0.207        0.0224      NA       
2 year        -0.000103     0.0000112    2.63e-20
3 UR           0.0000106    0.00000340   1.84e- 3
4 SDII        -0.000115     0.0000249    3.85e- 6
5 TASMAX       0.00000606   0.0000127    6.33e- 1
6 TASMIN       0.0000292    0.0000112    8.96e- 3
7 PREC_SUM    -0.000000212  0.000000868  8.07e- 1

Bioma Cerrado


# A tibble: 5 × 12
   year     mean       sd         var   min quant25 median  quant75    max     n  zero zero.perc
  <dbl>    <dbl>    <dbl>       <dbl> <dbl>   <dbl>  <dbl>    <dbl>  <dbl> <int> <int>     <dbl>
1  2001 0.000244 0.00112  0.00000125      0       0      0 0.000136 0.0224  1070   588     0.550
2  2002 0.000244 0.00100  0.00000101      0       0      0 0.000100 0.0169  1070   599     0.560
3  2003 0.000255 0.000856 0.000000732     0       0      0 0.000131 0.0135  1070   625     0.584
4  2004 0.000244 0.000839 0.000000703     0       0      0 0.000143 0.0129  1070   611     0.571
5  2005 0.000238 0.000771 0.000000595     0       0      0 0.000154 0.0110  1070   600     0.561

# A tibble: 7 × 4
  var            Estimate `Std. Error`   p.value
  <chr>             <dbl>        <dbl>     <dbl>
1 (Intercept)  0.0160      0.0122      NA       
2 year        -0.00000841  0.00000614   0.171   
3 UR           0.00000803  0.00000211   0.000143
4 SDII        -0.0000241   0.0000134    0.0736  
5 TASMAX       0.00000963  0.00000748   0.198   
6 TASMIN       0.0000107   0.00000643   0.0966  
7 PREC_SUM     0.00000101  0.000000460  0.0288  

Bioma Mata Atlantica


# A tibble: 5 × 12
   year      mean       sd         var   min quant25 median quant75     max     n  zero zero.perc
  <dbl>     <dbl>    <dbl>       <dbl> <dbl>   <dbl>  <dbl>   <dbl>   <dbl> <int> <int>     <dbl>
1  2001 0.000144  0.000736 0.000000541     0       0      0       0 0.0182   2751  2074     0.754
2  2002 0.000118  0.000590 0.000000348     0       0      0       0 0.0112   2751  2126     0.773
3  2003 0.000105  0.000802 0.000000643     0       0      0       0 0.0255   2751  2163     0.786
4  2004 0.0000806 0.000388 0.000000151     0       0      0       0 0.0109   2751  2166     0.787
5  2005 0.0000768 0.000331 0.000000110     0       0      0       0 0.00859  2751  2182     0.793

# A tibble: 7 × 4
  var             Estimate `Std. Error`   p.value
  <chr>              <dbl>        <dbl>     <dbl>
1 (Intercept)  0.0534       0.00679     NA       
2 year        -0.0000270    0.00000340   1.99e-15
3 UR           0.00000572   0.00000105   5.75e- 8
4 SDII        -0.0000226    0.00000815   5.52e- 3
5 TASMAX       0.0000189    0.00000360   1.65e- 7
6 TASMIN       0.00000592   0.00000287   3.95e- 2
7 PREC_SUM    -0.000000543  0.000000237  2.22e- 2

Bioma Pampa


# A tibble: 5 × 12
   year        mean         sd      var   min quant25 median quant75       max     n  zero zero.perc
  <dbl>       <dbl>      <dbl>    <dbl> <dbl>   <dbl>  <dbl>   <dbl>     <dbl> <int> <int>     <dbl>
1  2001 0.000000935 0.00000896 8.02e-11     0       0      0       0 0.0000957   118   114     0.966
2  2002 0.00000198  0.0000127  1.60e-10     0       0      0       0 0.000105    118   114     0.966
3  2003 0.00000490  0.0000523  2.73e- 9     0       0      0       0 0.000568    118   116     0.983
4  2004 0.00000334  0.0000258  6.65e-10     0       0      0       0 0.000265    118   115     0.975
5  2005 0.00000135  0.00000898 8.07e-11     0       0      0       0 0.0000781   118   114     0.966

# A tibble: 7 × 4
  var             Estimate `Std. Error`   p.value
  <chr>              <dbl>        <dbl>     <dbl>
1 (Intercept)  0.000911    0.00179      NA       
2 year        -0.000000495 0.000000898   0.581   
3 UR          -0.000000303 0.000000423   0.475   
4 SDII         0.00000410  0.00000311    0.188   
5 TASMAX       0.00000190  0.00000120    0.114   
6 TASMIN       0.000000356 0.000000739   0.630   
7 PREC_SUM     0.000000296 0.0000000781  0.000166

Referência


A análise estatística foi realizada no ambiente de computação estatística R (R Core Team, 2022). Os principais pacotes R utilizados foram o {dplyr} (Wickham et al., 2022), {tidyr} (Wickham & Girlich, 2022), {ggplot2} (Wickham, 2016), {patckwork} (Pedersen, 2020), {geobr} (Pereira & Gonçalves, 2022), {rlang} (Henry & Wickham, 2022), {purrr} (Henry & Wickham, 2020), {psych} (Revelle, 2022), {lme4} (Bates et al., 2015), e {lmerTest} (Kuznetsova et al., 2017).

R Core Team (2022). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/

Wickham, H., François, R., Henry, L., Müller, K. (2022). dplyr: A Grammar of Data Manipulation. R package version 1.0.9. https://CRAN.R-project.org/package=dplyr

Wickham, H., Girlich, M. (2022). tidyr: Tidy Messy Data. R package version 1.2.0, https://CRAN.R-project.org/package=tidyr

Wickham, H. (2016). ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York

Pedersen, T. L. (2020). patchwork: The Composer of Plots. R package version 1.1.1. https://CRAN.R-project.org/package=patchwork

Pereira, R. H. M., Gonçalves, C. N. (2022). geobr: Download Official Spatial Data Sets of Brazil. R package version 1.6.5999, <https://github.com/ipeaGIT/geobr

Henry, L., Wickham, H. (2022). rlang: Functions for Base Types and Core R and ‘Tidyverse’ Features. R package version 1.0.3, https://CRAN.R-project.org/package=rlang

Henry, L., Wickham, H. (2020). purrr: Functional Programming Tools. R package version 0.3.4, https://CRAN.R-project.org/package=purrr

Revelle, W. (2022). psych: Procedures for Personality and Psychological Research, Northwestern University, Evanston, Illinois, USA, R package version 2.2.3, https://CRAN.R-project.org/package=psych

Bates, D., Maechler, M., Bolker, B., Walker, S. (2015). Fitting Linear Mixed-Effects Models Using lme4. Journal of Statistical Software, 67(1), 1-48. doi:10.18637/jss.v067.i01

Kuznetsova, A., Brockhoff, P. B., Christensen, R. H. B. (2017). lmerTest Package: Tests in Linear Mixed Effects Models. Journal of Statistical Software, 82(13), 1-26. doi:10.18637/jss.v082.i13